Size and mass of very large stars: Most massive example, the blue Pistol Star (150 $\begin{smallmatrix}M_\odot\end{smallmatrix}$). Others are Rho Cassiopeiae (40 $\begin{smallmatrix}M_\odot\end{smallmatrix}$), Betelgeuse (20 $\begin{smallmatrix}M_\odot\end{smallmatrix}$), and VY Canis Majoris (30-40 $\begin{smallmatrix}M_\odot\end{smallmatrix}$). (The Sun (1 $\begin{smallmatrix}M_\odot\end{smallmatrix}$) which is not visible in this thumbnail is included to illustrate the scale of example stars. Earth's orbit (grey), Jupiter's orbit (red), and Neptune's orbit (blue) are also given.)

The solar mass ($\begin{smallmatrix}M_\odot\end{smallmatrix}$) is a standard unit of mass in astronomy that is used to indicate the masses of other stars, as well as clusters, nebulae and galaxies. It is equal to the mass of the Sun, about two nonillion kilograms:

$M_{\odot}=( 1.98855\ \pm\ 0.00025 )\ \times10^{30}\hbox{ kg}$[1][2]

The above mass is about 332,946 times the mass of the Earth or 1,048 times the mass of Jupiter.

Because the Earth follows an elliptical orbit around the Sun, its solar mass can be computed from the equation for the orbital period of a small body orbiting a central mass.[3] Based upon the length of the year, the distance from the Earth to the Sun (an astronomical unit or AU), and the gravitational constant (G), the mass of the Sun is given by:

$M_\odot=\frac{4 \pi^2 \times (1\ {\rm AU})^3}{G\times(1\ {\rm year})^2}$.

The value of the gravitational constant was derived from measurements that were made by Henry Cavendish in 1798 from his using a torsion balance. The value he obtained differed only by about 1% from the modern value.[4] The diurnal parallax of the Sun was accurately measured during the transits of Venus in 1761 and 1769,[5] yielding a value of 9″ (compared to the present 1976 value of 8.794148″). If we know the value of the diurnal parallax, we can determine the distance to the Sun from the geometry of the Earth.[6]

The first person to estimate the mass of the Sun was Isaac Newton. In his work Principia, he estimated that the ratio of the mass of the Earth to the Sun was about 1/28,700. Then later he determined that his value was based upon a faulty value for the solar parallax, which he had used to estimate the distance to the Sun (1 AU). So, he revised his result to obtain a ratio of 1/169,282 in the third edition of the Principia. The current estimated value for the solar parallax is smaller still, giving us a mass ratio of 1/332,946.[7]

As a unit of measurement, the solar mass came into use before the AU and the gravitational constant were precisely measured. This is because the determination of the relative mass of another planet in the Solar System or of a binary star in units of solar masses does not depend on these poorly known constants. So it was useful to express these masses in units of solar masses (see Gaussian gravitational constant).

The mass of the Sun changes slowly, compared to the lifetime of the Sun. Mass is lost due to two main processes in nearly equal amounts. First, in the Sun's core hydrogen is converted into helium by nuclear fusion, in particular the pp chain. Thereby mass is converted to energy in correspondence to the mass–energy equivalence. This energy is eventually radiated away by the Sun. The second process is the solar wind, which is the ejection of mainly protons and electrons to outer space. The actual net mass of the Sun since it reached the main sequence remains uncertain. The early Sun had much higher mass loss rates than at present, so, realistically, it may have lost anywhere from 1–7% of its total mass over the course of its main sequence lifetime.[8] The Sun also gains mass when foreign bodies such as asteroids and comets crash into it. Because the Sun already holds 99.86% of the Solar System's total mass, foreign body impacts are not expected to offset its loss of mass by the two aforementioned processes.

## Related units

One Solar mass, M, can be converted to related units:

It is also frequently useful in general relativity to express mass in units of length or time.

$M_\odot \frac{G}{c^2} \approx 1.48~\mathrm{km};\ \ M_\odot \frac{G}{c^3} \approx 4.93~ \mathrm{\mu s}$

## References

1. ^ 2013 Astronomical Constants http://asa.usno.navy.mil/SecK/2013/Astronomical_Constants_2013.pdf
2. ^ NIST CODATA http://physics.nist.gov/cgi-bin/cuu/Value?bg
3. ^ Harwit, Martin (1998), Astrophysical concepts, Astronomy and astrophysics library (3 ed.), Springer, pp. 72, 75, ISBN 0-387-94943-7
4. ^ Holton, Gerald James; Brush, Stephen G. (2001). Physics, the human adventure: from Copernicus to Einstein and beyond (3rd ed.). Rutgers University Press. p. 137. ISBN 0-8135-2908-5.
5. ^ Pecker, Jean Claude; Kaufman, Susan (2001). Understanding the heavens: thirty centuries of astronomical ideas from ancient thinking to modern cosmology. Springer. pp. 291–291. ISBN 3-540-63198-4.
6. ^ Barbieri, Cesare (2007). Fundamentals of astronomy. CRC Press. pp. 132–140. ISBN 0-7503-0886-9.
7. ^ Leverington, David (2003). Babylon to Voyager and beyond: a history of planetary astronomy. Cambridge University Press. p. 126. ISBN 0-521-80840-5.
8. ^ Sackmann, I.-Juliana; Boothroyd, Arnold I. (February 2003), "Our Sun. V. A Bright Young Sun Consistent with Helioseismology and Warm Temperatures on Ancient Earth and Mars", The Astrophysical Journal 583 (2): 1024–1039, arXiv:astro-ph/0210128, Bibcode:2003ApJ...583.1024S, doi:10.1086/345408